A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Infinitesimal Generator of Ito Diffusion Process

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The following is an excerpt from wikipedia My question is on how to derive this operator? It looks very ...
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Compound Poisson Process property:$\mathbb{P}(\sum^{N_{t_4}-N_{t_3}}_{i=1}J_i \leq n)=\mathbb{P}(\sum^{N_{t_4}}_{i=N_{t_3}+1}J_i \leq n)$

I am trying to demostrate that the Compound Poisson Process has independent increments, and I have a problen because I have to use that: :$$\mathbb{P}(\sum^{N_{t_4}-N_{t_3}}_{i=1}J_i \leq ...
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Is the Martingale property still true for $\xi$ not necessarily $C^1$?

Denote $$M^n(t) = f(t, \alpha(t))\exp \bigg\{-\int_0^t g(u, \alpha (u)) \, du - \int_0^t h(u, \alpha(u)) \, d\xi(u)\bigg\}$$ Here $\xi: [0,\infty) \times \Omega \to \Bbb{R}$. If for each $\omega$ ...
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Understanding Quadratic Variation

I think part of the trouble a lot of people (or at least me personally) have with making the jump from calculus to stochastic calculus is the notion of quadratic variation. It doesn't have as much ...
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Reference request for this topics

I need a good reference to learn these topics Markov Chains in discrete time.    1.1. Classification of states, recurrence notions of transience.    1.2. Stationary measure.    1.3. ...
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Non existence of probabilty measures.

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ ist standard Wiener. This solution is ...
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1answer
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how do I parametrise a stochastic matrix

I have a matrix $\mathbf{t}$ that maps one $d$ dimensional probability distribution to another $\mathbf{t}^T x = q$, i.e. with $\sum\limits_i t_{ij} x_i = q_j$ and $\sum\limits_j t_{ij} = 1$ $\forall$ ...
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1answer
18 views

If $(B_t)_{t\ge 0}$ is a Brownian motion and $\tau$ is a stopping time, then the stopped process $(B_{\min(\tau,t)})_{t\ge 0}$ is integrable

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$. By definition $B_t$ is normally distributed with mean $0$ and variance $t$. Now, let ...
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9 views

Bounded L2 increments for an Ornstein Uhlenbeck type process

Let $Z$ be an increasing Levy process (i.e. a subordinator). Let $\lambda>0$ and consider the Ornstein Uhlennbeck type SDE $$ d V_t = - \lambda V_t dt + d Z_{\lambda t } $$ where the integral can ...
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Probability that a birth--death process crosses level $n$ in $(0,T)$

This question is inspired by this question. Jobs arriving according to a Poisson process with rate $\lambda$. Jobs stay in the system for a fixed amount of time $d$ and depart thereafter. Let $X(t)$ ...
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20 views

Proving that the Poisson compound process has independet increments

Let $X_t=\sum_{i=1}^{N_t}J_i$ be a compound Poisson Process, where $J_i$ are independent and equidistributed. I have to prove that for every $0<t_1<t_2 \leq t_3<t_4$ : $X_{t_4}-X_{t_3}$ is ...
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11 views

Optional stopping/sampling for right-continuous supermartingales

Let $\mathbb{F}$ be a filtration $(X_t)_{t\ge 0}$ be a right-continuous $\mathbb{F}$-supermartingale $\sigma,\tau$ be bounded $\mathbb{F}$-stopping times with $\sigma\le \tau$ and ...
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1answer
26 views

Show that $W$ is a Gaussian process

I have the following problem: I want to prove that the vector $(W(1_{[t_0,t_1]}),...,W(1_{[t_{n-1},t_n]}))$ is normally distributed with mean $0$ and covariance matrix ...
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1answer
31 views

Variation processes and strong solutions of stochastic differential equations

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$ $\tau$ be a $\mathbb{F}$-stopping time An $\mathbb{F}$-adapted, ...
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19 views

Power Spectral Density Approximation

Let $X_t$ be a zero-mean, stationary random process. Let $X_f$ be the Fourier transform of $X_t$; $X_f$ is also a random process, but as a function of $f$. Let us denote the power spectral density ...
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21 views

Recursive Variance

What will be the distribution or features about the following $x$? $x=\mu+\epsilon$ where $\epsilon\sim N(0,x^{-1})$. It seems interesting in econometrics if we allow $x$ being a time series and ...
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15 views

Infinitesimal Generator for Stochastic Processes

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The infinitesimal generator $LV(x)$ is defined by: $$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) ...
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16 views

Transient discrete time Markov chain on integers: can direction of flow be proven?

I'm not very familiar with the theory of Markov chains, and I'd like to learn how complicated the following problem actually is. Let there be a discrete time Markov chain on $\mathbb{Z}$, where the ...
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1answer
41 views

Doob decomposition of $|S_n|$ where $S_n$ is simple random walk.

Let $X_n$, $n\geqslant 1$ be iid Rademacher random variables, i.e. $X_1$ takes values $\pm 1$ each with probability $\frac12$. Define $S_0=0$ and $S_n=\sum_{i=0}^n X_i$, and $\mathcal F_n = ...
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Conditioning on Brownian motion

I was reading on conditional probability with respect to a partition of a sample space, and I came across the following example: Let $(N_t:t\geq0)$ be the Poisson process. Given fixed times $0\leq ...
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19 views

Markov Chain with dependence between users

I am looking for a Markov Chain model that describes the following problem. I have $N$ indifferent users in the system, each of them has three states: $A$, $B$, $C$, and I know the transition ...
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1answer
25 views

Infinitesimal Generator of Poisson process

I would like to compute the infinitesimal generator of a Poisson process $N$ with intensity $\lambda$. So I can write: $$\mathbb{E}[\ f(N_{t+s})-f(N_s)\ |\ \mathcal{F_t^0} \ ] = \mathbb{E}[\ ...
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1answer
34 views

If $(X_t,t\in I)$ is a process with values in $(E,\mathcal{E})$, are $\sigma(X_t,t\in I)$ and $\sigma(X)=X^{-1}(\mathcal{E}^{\otimes I})$ equal?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $E$ be a Polish space and $\mathcal{E}$ be the Borel $\sigma$-algebra on $E$ $I$ be an index set $X_t$ be a random variable on ...
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1answer
13 views

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Proove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, ...
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2answers
39 views

Exchanging expectation and limits

Exchanging expectation and limits I have a stochastic process, ${b_t} \, (t=0, 1, 2, \ldots)$, which follows a random walk. Specifically, ${b_0} = 0$ and for $t$ greater than zero, $\displaystyle ...
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Doob's submartingale stopping theorem in the context of the submartingale problem

Let $$X^\omega_f (t, w) = f(w(t)) - f(w(t \wedge \tau)) - \frac{1}{2} \int_{t \wedge \tau}^t \Delta f(w(s))\, ds$$ be a $P^\tau_\omega$-submartingale. 1) Why Doob's submartingale stopping theorem ...
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1answer
24 views

Applying the Multivariate Ito Formula

I want to show that the stochastic process $$ S_t^i = S_0^i \exp\left( \int_0^t \left(\mu_s^i - \frac{1}{2} \sum_{j=1}^m (\sigma_s)^{ij} \right)^2 d s + \sum_{j=1}^m \sigma_t^{ij} S_t^i dW_t^j ...
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Aggregate arrivals from a renewal process

This is a follow-up question of the question "Aggregate arrivals from a Poisson Process". The inter-arrival time of a renewal process, t, conforms to a general distribution, denoted by PDF $f(t)$. ...
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1answer
27 views

Aggregate arrivals from a Poisson Process

The inter-arrival time of a Poisson Process, $t$, conforms to the exponential distribution, so the probability density function for $t$ is $f(t)=λe^{−λt},~t>0$. ($λ$ is the arrival rate of the ...
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Simulation of interacting Ornstein-Uhlenbeck processes

I would like to simulate the following system of interacting OU processes on $[0,T]$: $$dX_t^1=(X_t^2-X_t^1)\,dt+\sigma_1 \,dW_t^1,\quad X_0^1=x_1$$ $$dX_t^2=(X_t^1-X_t^2)\,dt+\sigma_2 ...
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35 views

Mean-Square Ergodicity of Certain Quantities?

I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way. Is it possible ...
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1answer
24 views

Working with the random variable $\log X$ instead of $X$

Suppose I have a positive stochastic process $X_t$. I'd like to compute certain properties about $X_t$, but suppose I can't and instead I can compute properties about $\log(X_t)$. Can I say anything ...
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Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
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process stochastics and branching process [duplicate]

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
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1answer
27 views

Is a martingale with bounded variance therefore bounded in $L^2$?

If a martingale $W_n$ has bounded variance, does this mean that $W_n$ is automatically bounded in $L^2$? I feel like this ought to be obvious but I don't see how to prove it and I haven't been able to ...
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8 views

discontinuous Gaussian field

I am trying to build an example of a discontinuous Gaussian field. The simplest I could come up with is the following: Let $Y,Z$ be two independent brownian motions on $[0,1]$, and $T$ a uniform ...
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22 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
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1answer
57 views

Conditional probability branching process

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
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$\sup B_t$ has the same distribution as $\sup C_t$ for two brownian motions $B_t, C_t$

Let $(B_t)_{t \ge 0}$ and $(C_t)_{t \ge 0}$ be two standardized brownian motions. Now why is $\sup_{t \ge 0} B_t$ distributed same as $\sup_{t \ge 0} C_t$? This is a result we assumed as trivial ...
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1answer
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$W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$?

This is a sample question for the actuarial exam MFE. Let $Z(t)$ be a standard Brownian motion. Let $W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$? The only thing I know is Ito's Lemma. So I ...
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24 views

FFT Hyperbolic Distribution R

This is my first posting so forgive me if it is not 100% in line with this forum's best practices. I am completing an analysis using ICA as the decomposition technique. I am keeping 4 of the 10 ...
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24 views

Simple Markov property

I want to prove the simple Markov property but I come to a point where I do not see how to conclude. I want to prove $\mathbb{E}_\nu[Z\circ\Theta_t\mid ...
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25 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
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Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...
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19 views

Represent stochastic process as conditional expectation

I try to reduce my problem to the following question, which is stated rather sloppy (without possibly necessary additional conditions). Let $Y_t$ be a real stochastic process for $t \in [0, T]$ and ...
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154 views
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Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$

I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way. Assumptions: Consider a ...
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24 views

Prove that an integral is zero (from Gardiner's Handbook of stochastic methods)

I have troubles in one proof of the book Handbook of stochastic methods by Gardiner. In the paragraph 3.7.3 he writes this integral $\sum_i\int d\vec x \frac{\partial}{\partial ...
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1answer
27 views

Central limit theorem - generalizations [on hold]

I am looking for some generalizations for the Central limit theorem in the following sense: Let $\phi$ be a function on the natural numbers, under what conditions on $\phi$ $ ...
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1answer
21 views

construct a martingale process from any process [closed]

If ${Z_n, n \geq 0}$ is any sequence of integrable random variables, then ${\sum_{i=1}^{n}[Z_i-E(Z_i|Z_{i-1},...,Z_1)]}$ is a martingale relative to the sequence of $\sigma$-fields generated by $Z_i$, ...
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1answer
28 views

How to get the basis of $L^2[0,1]$ from the basis of $L^2[0,2]$

Is there any way to derive orthonormal basis of $L^2[0,1]$ from the orthonormal basis of $L^2[0,2]$? Here $L^2[0,2]$: is space of square integrable centered stochastic process on $\Omega\times[0,2]$, ...